Can I snap an objects so their edges line up?
I not sure what problems you have had in the past with blenders snapping in the past but I'd personal use the snapping options here as I believe it'd be the fastest and most simple way to get it done. I apologize in advance if you were tying to avoid the snap tool completely for some reason.

My solution:

Set the snap element to vertex, by using the button next to the snap button on the 3D views header bar OR press ctrl+shift+tab to bring the snapping options menu up by it's shortcut. 



Select the cube. Press G to enter grab/move mode. Hold ctrl down and point the plains lower right vertex and click to confirm the move.



If it all went right the cube should be perfectly aligned to the corner. If you duplicate it then hold ctrl and point to another corner it should make a new copy and snap it to whatever corner you point to.

Note on How snapping works in blender:

When snapping two elements together blender will try and snap the closets points together, from where The moved object starts. 

So if your cube is above and to the left of the vertex you are trying to snap to, The cubes bottom right vertex will be the closest point and that will be the part that snaps into position.

I hope I've explained this all clearly and that it helps you out! good luck!